Discrete Mathematics Lecture Notes

Questions and Answers

What does the matrix MR represent?

A relation from set B to set A

A relation from set A to set B (correct)

A function of set A

A relation between two different sets

Which of the following ordered pairs is included in the set R derived from matrix MR?

(a2, b1)

(a1, b4) (correct)

(a3, b2)

(a2, b4)

What is the relationship between MR and MR−1?

MR is the transpose of MR−1

MR−1 is the transpose of MR (correct)

MR is the inverse of MR−1

MR and MR−1 are equal

In the context of a directed graph representation, what do the arrows represent?

The direction of the relation Signup and view all the answers

Given the relation from A to B, how would you represent no relation from an element a1 to b2 in the matrix MR?

0 Signup and view all the answers

How many elements would the digraph of relation R have based on the set A = {a1, a2, a3}?

3 vertices Signup and view all the answers

What is the structure of MR if the relation contains no connections?

All values are 0 Signup and view all the answers

What type of relation is represented if there are arrows going from one vertex to multiple vertices in a digraph?

One-to-many relation Signup and view all the answers

What is the correct notation to denote that an element 'a' is part of set A?

a ∈ A Signup and view all the answers

How are sets denoted in mathematical notation?

With upper case letters Signup and view all the answers

In set theory, what is the significance of the order of elements in a set?

The order is irrelevant when defining a set Signup and view all the answers

Which notation is used to indicate that a certain condition is met by members of a set?

Predicate Notation Signup and view all the answers

Which of the following correctly represents a set of prime numbers less than 10?

{2, 3, 5, 7} Signup and view all the answers

What is the correct description of the symbols '∈' and '∉'?

'∈' indicates membership and '∉' indicates non-membership Signup and view all the answers

In the context of set theory, what does the universal set refer to?

A set containing all elements in a particular context Signup and view all the answers

What are the correct symbols for the set containing only the even numbers?

{2n | n ∈ Z} Signup and view all the answers

What is the domain of the relation R defined from A = {1, 2, 3, 4} to B = {x, y, z}?

{1, 3, 4} Signup and view all the answers

What is the range of the relation R defined from A = {1, 2, 3, 4} to B = {x, y, z}?

{y, z, x} Signup and view all the answers

How is the relation R represented in matrix form?

MR = [[1, 1, 0], [0, 1, 1], [1, 0, 0], [0, 1, 0]] Signup and view all the answers

What is the inverse relation R −1 of R, given R = {(1, y), (1, z), (3, y), (4, x), (4, z)}?

(y, 1), (z, 1), (y, 3), (x, 4), (z, 4) Signup and view all the answers

Which of the following correctly describes the notation MR for a relation?

MR = [[mij]] represents a matrix where mij = 0 if (ai, bj) ∉ R Signup and view all the answers

If the matrix MR of a relation contains only 1s, what does this imply about the relation?

Every element of the domain relates to every element of the codomain. Signup and view all the answers

Which of the following sequences is a solution to the recurrence relation $a_n = 8a_{n-1} - 16a_{n-2}$?

$a_n = 1$ Signup and view all the answers

What initial conditions would define the recurrence relation for deeper population growth in this model?

$d_0 = 1000$, $d_1 = 1100$ Signup and view all the answers

How much will be in the account after 100 years with a 9% interest compounded annually on an initial deposit of $1000?

$1000(1.09)^{100}$ Signup and view all the answers

What will be the world's population in 2050 if it is currently 6 billion and grows at a rate of 1.3% per year?

$6(1.013)^{30}$ Signup and view all the answers

Which of the following is not a characteristic of linear homogeneous recurrence relations?

They can include non-homogeneous terms. Signup and view all the answers

What is the first term of the deeper population growth after one time period given the initial condition?

1100 Signup and view all the answers

For the growth rate derived from a 10% increase per time period, which equation describes the population at time n?

$d_n = (1.1)^n d_0$ Signup and view all the answers

Which expression correctly represents the amount in the account after n years with a principal $P$ at an interest rate $r$?

$A = P(1 + r)^n$ Signup and view all the answers

Is the relation R = {(a, a), (b, b), (c, b), (d, d)} transitive?

Yes, it is transitive. Signup and view all the answers

Which of the following properties must hold true for a relation to be an equivalence relation?

Reflexive, symmetric, and transitive. Signup and view all the answers

What can be inferred if a relation R is reflexive and antisymmetric but not transitive?

R is a partial order. Signup and view all the answers

Given the relation R = {(a, a), (b, c), (c, b), (d, d)}, is R transitive?

No, because (b, b) is not present. Signup and view all the answers

In the set {1, 2, 3, 4}, which relation can be reflexive, symmetric, and not transitive?

R = {(1,1), (2,2), (3,3), (1,2), (2,1), (3,4)} Signup and view all the answers

Which task must necessarily be completed before turning on the flash unit?

Remove the lens cap. Signup and view all the answers

For the relation R = {(a, a), (b, b), (c, b), (d, d)}, which property does it not satisfy?

It is symmetric. Signup and view all the answers

What characteristic is not required for a partial order?

Symmetry. Signup and view all the answers

Study Notes

Expected Learning Outcomes

Ability to explain, solve and apply logical relationships.
Competence in using sets and relations to solve problems.
Familiarity with properties of numbers, relations, and functions in problem-solving.
Understanding and applying inclusion and exclusion principles.
Skills to resolve problems related to elementary graph theory.

Set Theory

A set is a well-defined collection of objects or numbers, denoted by uppercase letters.
Elements of a set are denoted by lowercase letters, represented in curly brackets (e.g., A = {a, b, c}).
Membership of an element in a set is indicated using ∈ (e.g., a ∈ A) or ∉ (e.g., x ∉ A).

Specifying a Set

List Notation: Members are listed in curly brackets (e.g., C = {Kshs, USD, Euro, Pound}). Order of elements is irrelevant.
Predicate Notation: Specifies sets based on shared conditions, useful when complete membership is unknown.

Representation of Relations

Relations between two sets A and B can be expressed using a matrix, where the entry is 1 if an ordered pair belongs to the relation and 0 if not.
For sets A and B, the matrix ( M_R ) is an ( m \times n ) grid where ( m ) and ( n ) are the number of elements in sets A and B, respectively.

Matrix Representation Example

Given A = {1, 2, 3} and B = {r, s}, the relation R defined by ( R = {(1, r), (2, s), (3, r)} ) has the matrix representation: [ M_R = \begin{pmatrix} 1 & 0 \ 0 & 1 \ 1 & 0 \end{pmatrix} ]

Undirected Graph Representation

Relations can be depicted as directed graphs (digraphs) by drawing circles for each element and using arrows to show connections based on the relation.

Transitive Relations

A relation is transitive if whenever it contains pairs (a, b) and (b, c), it must also contain (a, c).
Examples provided to discern transitive properties in relations.

Equivalence Relations

An equivalence relation is reflexive (every element relates to itself), symmetric (if a relates to b, then b relates to a), and transitive.
A partial order is reflexive, antisymmetric (if a relates to b and b relates to a, then a must be equal to b), and transitive.

Exercises

Tasks include providing specific examples of relations based on defined properties (reflexive, symmetric, transitive) and interpreting digraphs.

Linear Homogeneous Recurrence Relation

Defined as ( a_n = c_1 a_{n-1} + c_2 a_{n-2} + \ldots + c_k a_{n-k} ) for constants ( c_i ).
Examples illustrate how to formulate a recurrence relation based on a given scenario, such as population change over time.

Applications of Recurrence Relations

Practice problems set for tasks such as population growth, interest accumulation, and scheduling, demonstrating real-world scenarios where recurrence relations apply.
Example of population growth initiates with a given amount and growth percentage, establishing foundational computational methods.

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Description

Explore key concepts and applications in Discrete Mathematics with these comprehensive lecture notes. Learn about logical relationships, sets, relations, and properties of numbers essential for problem-solving in the field. Perfect for students aiming to deepen their understanding of discrete structures and their practical applications.